**2. Problem Formulation and Coordinate System**

The geometry of the problem (see Figure 1) is based on the MHD effect for axisymmetric Marangoni convective, incompressible, steady and laminar flow utilizing the electrically conducting Casson nanoliquid model. Marangoni convective flow is caused due to concentration and temperature gradients on surfaces generated by surface tension. A uniform magnetic field is applied in such a way that it makes an angle *α*1 in non-vertical direction. The cylindrical coordinates system is considered along and normal to the interface of flow problem. Concentration and temperature interfaces of the flow structure are altered at the surface of the disk. The analysis of heat transfer is examined through Joule heating and viscous dissipation. The formulated governing equations for the MHD effect on Marangoni convective flow structure are given as (see, for example, [4–15]):

$$
\frac{\partial \vec{u}}{\partial \vec{r}} + \frac{\partial \vec{w}}{\partial \vec{z}} + \frac{\vec{u}}{\vec{r}} = 0,\tag{1}
$$

$$i\hbar \frac{\partial \vec{u}}{\partial \vec{r}} + i\tilde{v} \frac{\partial \vec{u}}{\partial \vec{z}} = \frac{\mu}{\rho} \left( 1 + \frac{1}{\beta\_1} \right) \frac{\partial^2 \vec{u}}{\partial \vec{z}^2} - \frac{\sigma B\_0^2}{\rho} \sin^2 \alpha\_1 \tilde{u},\tag{2}$$

$$\begin{split} \boldsymbol{\tilde{u}} \frac{\partial \boldsymbol{\Upsilon}}{\partial \boldsymbol{\tilde{r}}} + \boldsymbol{\tilde{w}} \frac{\partial \boldsymbol{\mathcal{T}}}{\partial \boldsymbol{\tilde{z}}} &= \quad \frac{\boldsymbol{k}}{\rho \boldsymbol{c}\_{\omega}} \frac{\partial^{2} \boldsymbol{\mathcal{T}}}{\partial \boldsymbol{\tilde{z}}^{2}} + \boldsymbol{\tau} \frac{\boldsymbol{D}\_{\boldsymbol{\mathcal{T}}}}{\boldsymbol{\tilde{T}}\_{\infty}} \left( \frac{\partial \boldsymbol{\mathcal{T}}}{\partial \boldsymbol{\tilde{z}}} \right)^{2} + \boldsymbol{\tau} \boldsymbol{D}\_{\boldsymbol{B}} \frac{\partial \boldsymbol{\tilde{C}}}{\partial \boldsymbol{\tilde{z}}} \frac{\partial \boldsymbol{\mathcal{T}}}{\partial \boldsymbol{\tilde{z}}} + \frac{1}{\rho \boldsymbol{c}\_{\omega}} \boldsymbol{Q}\_{1} (\boldsymbol{\mathcal{T}} - \boldsymbol{\mathcal{T}}\_{\infty}) \\ &+ \frac{\mu}{\rho \boldsymbol{c}\_{\omega}} \left( 1 + \frac{1}{\beta\_{1}} \right) \left( \frac{\partial \boldsymbol{\tilde{u}}}{\partial \boldsymbol{\tilde{z}}} \right)^{2} + \frac{\sigma B\_{0}^{2}}{\rho \boldsymbol{c}\_{\omega}} \sin^{2}{\boldsymbol{\alpha}\_{1} \boldsymbol{\tilde{u}}} \boldsymbol{\alpha}\_{1} \boldsymbol{\tilde{u}}^{2}, \end{split} \tag{3}$$

$$
\hbar \frac{\partial \tilde{\mathcal{C}}}{\partial \tilde{r}} + \imath \tilde{\nu} \frac{\partial \tilde{\mathcal{C}}}{\partial \tilde{z}} = D\_B \frac{\partial^2 \tilde{\mathcal{C}}}{\partial \tilde{z}^2} + \frac{D\_T}{\Upsilon\_{\infty}} \frac{\partial^2 \mathcal{T}}{\partial \tilde{z}^2}. \tag{4}
$$

In Equations (1)–(4), *ρ* characterizes fluid density; *μ* signifies dynamic viscosity; *β* indicates parameter of Casson fluid; *σ* symbolizes surface tension; *C* ˜ and *T* ˜ represent fluid concentration and temperature, respectively; *C* ˜ ∞ and *T* ˜ ∞ characterize fluid ambient concentration and temperature far away from the surface, respectively; *τ* shows shear stress; *DB* is the coefficient of Brownian diffusion; *k* indicates coefficient of absorption; *cω* denotes specific heat; *DT*˜ characterizes coefficient of thermophoretic diffusion; *Q*1 represents heat source sink coefficient; and *α*1 signifies angle of inclination.

**Figure 1.** Physical diagram of the flow model.

The subjected boundary conditions are (see, for example, [10]):

$$
\begin{array}{rcl}
\mu \left( 1 + \frac{1}{\beta\_1} \right) \left. \frac{\partial \vec{u}}{\partial \vec{z}} \right|\_{\underline{z} = 0} &=& \left. \frac{\partial \sigma}{\partial \vec{T}} \frac{\partial \vec{T}}{\partial \vec{r}} \right|\_{\underline{z} = 0} + \left. \frac{\partial \sigma}{\partial \vec{C}} \frac{\partial \vec{C}}{\partial \vec{r}} \right|\_{\underline{z} = 0}, & \left. \vec{w} \right|\_{\underline{z} = 0} = 0, \\
\left. \vec{T} \right|\_{\underline{z} = 0} &=& \left. \vec{T} \infty + A \vec{r}^2 \phi, \left. \vec{C} \right|\_{\underline{z} = 0} = \vec{\mathcal{C}} \infty + B \vec{r}^2 \zeta, \\
\left. \vec{u} \right|\_{\underline{z} \to \infty} & \longrightarrow \left. 0, \left. \mathcal{T} \right|\_{\underline{z} \to \infty} \longrightarrow \mathcal{T}\_{\infty}, \left. \mathcal{C} \right|\_{\underline{z} \to \infty} \longrightarrow \mathcal{C}\_{\infty}.
\end{array}
\right.
\end{array}
$$

*Mathematics* **2019**, *7*, 1087

The suitable transformations incorporated in the proposed flow structure are (see, for example, [6]):

$$\eta = \sqrt{\frac{b}{\nu}} \overline{z}, \text{ if } = b\overline{r} \, g'(\eta), \text{ } \overline{w} = -2\sqrt{b\nu}g(\eta), \text{ } \zeta = \frac{\vec{C} - \vec{\mathcal{C}\_{\infty}}}{\mathcal{C}\_{\mathcal{S}} - \mathcal{C}\_{\infty}}, \text{ } \phi = \frac{\vec{T} - \vec{T}\_{\infty}}{\mathcal{T}\_{\mathcal{S}} - \mathcal{T}\_{\infty}}. \tag{6}$$

Moreover, assumptions indicate that surface tension is a linear function of concentration and temperature, which may be represented as (see, for example, [10]):

$$
\sigma = \sigma\_0 - \gamma\_\uparrow (\mathcal{T} - \mathcal{T}\_\infty) - \gamma\_\downarrow (\mathcal{C} - \mathcal{C}\_\infty),
\tag{7}
$$

where *σ*0, *γT*˜ and *γC*˜ represent the positive constants. After incorporating the above-mentioned transformations into Equations (1)–(4), we obtain

$$\left(1+\frac{1}{\beta\_1}\right)\mathbf{g}^{\prime\prime\prime}+2\mathbf{g}\mathbf{g}^{\prime\prime}-(\mathbf{g}^{\prime})^2-M\_1^2\sin^2\alpha\_1\,\mathbf{g}^{\prime}=0,\tag{8}$$

$$\text{rg}(0) = 0, \ (1 + \frac{1}{\beta\_1})\text{g}'(0) = -2 \, M\_\text{d} \, (1 + R\_\text{d} \, \zeta(0)), \ \text{g}'(\infty) = 0,\tag{9}$$

$$\frac{1}{2}\boldsymbol{\phi}^{\prime\prime} + 2\operatorname{Pr}\_{\mathcal{S}}\boldsymbol{\uprho}^{\prime} + \operatorname{Pr}\operatorname{N}\_{\mathcal{I}}\boldsymbol{\uprho}^{\prime}\mathbb{I} + \operatorname{Pr}\operatorname{N}\_{\mathcal{I}}(\boldsymbol{\phi}^{\prime})^{2} + \operatorname{Pr}\operatorname{E}\left(1 + \frac{1}{\tilde{\rho}\_{1}}\right)(\boldsymbol{g}^{\prime\prime})^{2} + \operatorname{Pr}\operatorname{E}\boldsymbol{M}\_{1}^{2}\sin^{2}{a\_{1}}(\boldsymbol{g}^{\prime})^{2} + \operatorname{Pr}\operatorname{B}\_{1}\boldsymbol{\uprho} = \boldsymbol{0},\tag{10}$$

$$
\phi(0) = 1,\ \phi(\infty) = 0,\ \text{N}\_1^{\prime}\zeta(0) + \text{N}\_2^{\prime}\phi^{\prime}(0) = 0,\tag{11}
$$

$$\text{L}^{\prime\prime}\_{\text{}} + 2\text{L} \text{c g}^{\prime}\_{\text{}} + \frac{N\_2}{N\_1} \phi^{\prime\prime} = 0,\tag{12}$$

$$
\zeta(0) = 1,\ \zeta(\infty) \to 0. \tag{13}
$$

In Equations (8)–(13), *N*1 = *<sup>τ</sup>DB*(*C*˜*g*<sup>−</sup>*C*˜∞) *ν* indicates Brownian motion parameter, *N*2 = *<sup>τ</sup>DT*˜(*T*˜*g*<sup>−</sup>*T*˜∞) *<sup>ν</sup>T*˜∞ characterizes thermophoresis parameter, *M*1 = *σ<sup>B</sup>*20 8*ρb* shows magnetic number, *Ma* = *γT*˜ *μ*<sup>Ω</sup><sup>Ω</sup>*γ* signifies Marangoni number, *Ra* = *γT*˜ *B γT*˜ *A* shows Marangoni ratio parameter, *B*1 = *Qhρcω* represents heat source sink, Pr = *ρcωμ k* signifies Prandtl number, *Ec* = *u* ˘2 ∞ *<sup>c</sup>ω*(*T*˜*w*<sup>−</sup>*T*˜∞) indicates Eckert number, and *Le* = *νDB* denotes Lewis number. Additionally, *Nu* the local Nusselt number is given as

$$N\mu = \frac{-\vec{r}\left(\frac{\partial \vec{T}}{\partial \vec{z}}\right)\Big|\_{\substack{z=0\\k(\vec{T}\_{\infty}-\vec{T}\_{w})}}}{k(\vec{T}\_{\infty}-\vec{T}\_{w})},\tag{14}$$

and in dimensionless form becomes

$$R\_d^{-1/2}Nu = \frac{Nu}{\sqrt{R\_d}} = -\phi'(0),\tag{15}$$

where *Rd* = *u* ˜ *wr* ˜ *ν* is local Reynold's parameter.
